Generalized relative entropy

Definition

To motivate the definition of the ϵ -relative entropy D ϵ ( ρ | | σ ) , consider the information processing task of hypothesis testing. In hypothesis testing, we wish to devise a strategy to distinguish between two density operators ρ and σ . A strategy is a POVM with elements Q and I − Q . The probability that the strategy produces a correct guess on input ρ is given by Tr ⁡ ( ρ Q ) and the probability that it produces a wrong guess is given by Tr ⁡ ( σ Q ) . ϵ -relative entropy captures the minimum probability of error when the state is σ , given that the success probability for ρ is at least ϵ .

For ϵ ∈ ( 0 , 1 ) , the ϵ -relative entropy between two quantum states ρ and σ is defined as

From the definition, it is clear that D ϵ ( ρ | | σ ) ≥ 0 . This inequality is saturated if and only if ρ = σ , as shown below.

Relationship to the trace distance

Suppose the trace distance between two density operators ρ and σ is

For 0 < ϵ < 1 , it holds that

In particular, this implies the following analogue of the Pinsker inequality

Furthermore, the proposition implies that for any ϵ ∈ ( 0 , 1 ) , D ϵ ( ρ | | σ ) = 0 if and only if ρ = σ , inheriting this property from the trace distance. This result and its proof can be found in Dupuis et al.

Proof of inequality a)

Upper bound: Trace distance can be written as

This maximum is achieved when Q is the orthogonal projector onto the positive eigenspace of ρ − σ . For any POVM element Q we have

so that if Tr ⁡ ( Q ρ ) ≥ ϵ , we have

From the definition of the ϵ -relative entropy, we get

Lower bound: Let Q be the orthogonal projection onto the positive eigenspace of ρ − σ , and let Q ¯ be the following convex combination of I and Q :

where μ = ( 1 − ϵ ) Tr ⁡ ( Q ρ ) 1 − Tr ⁡ ( Q ρ )   .

This means

and thus

Moreover,

Using μ = ( 1 − ϵ + μ ) Tr ⁡ ( Q ρ ) , our choice of Q , and finally the definition of μ , we can re-write this as

Hence

Proof of inequality (b)

To derive this Pinsker-like inequality, observe that

Alternative proof of the Data Processing inequality

A fundamental property of von Neumann entropy is strong subadditivity. Let S ( σ ) denote the von Neumann entropy of the quantum state σ , and let ρ A B C be a quantum state on the tensor product Hilbert space H A ⊗ H B ⊗ H C . Strong subadditivity states that

where ρ A B , ρ B C , ρ B refer to the reduced density matrices on the spaces indicated by the subscripts. When re-written in terms of mutual information, this inequality has an intuitive interpretation; it states that the information content in a system cannot increase by the action of a local quantum operation on that system. In this form, it is better known as the data processing inequality, and is equivalent to the monotonicity of relative entropy under quantum operations:

for every CPTP map E , where S ( ω | | τ ) denotes the relative entropy of the quantum states ω , τ .

It is readily seen that ϵ -relative entropy also obeys monotonicity under quantum operations:

for any CPTP map E . To see this, suppose we have a POVM ( R , I − R ) to distinguish between E ( ρ ) and E ( σ ) such that ⟨ R , E ( ρ ) ⟩ = ⟨ E † ( R ) , ρ ⟩ ≥ ϵ . We construct a new POVM ( E † ( R ) , I − E † ( R ) ) to distinguish between ρ and σ . Since the adjoint of any CPTP map is also positive and unital, this is a valid POVM. Note that ⟨ R , E ( σ ) ⟩ = ⟨ E † ( R ) , σ ⟩ ≥ ⟨ Q , σ ⟩ , where ( Q , I − Q ) is the POVM that achieves D ϵ ( ρ | | σ ) . Not only is this interesting in itself, but it also gives us the following alternative method to prove the data processing inequality.

By the quantum analogue of the Stein lemma,

where the minimum is taken over 0 ≤ Q ≤ 1 such that Tr ⁡ ( Q ρ ⊗ n ) ≥ ϵ   .

Applying the data processing inequality to the states ρ ⊗ n and σ ⊗ n with the CPTP map E ⊗ n , we get

Dividing by n on either side and taking the limit as n → ∞ , we get the desired result.